The Italian mathematician Eugenio Calabi died on September 25, 2023 at the age of 100 in Beaumont, Bryn Mawr (USA). This year there had been numerous tributes throughout the world, to celebrate his impressive legacy and his important contributions to geometry. It is unusual that the centenary of an important mathematician, with more than 70 years of scientific heritage, and three generations of descendants, is celebrated under his watchful eye – this is what happened at one of the conferences, held in Hefei (China) -.
Born in Milan, Italy in May 1923, Calabi moved to the United States with his family at a young age. He completed his studies at the Massachusetts Institute of Technology, funded by a prestigious Putnam scholarship, which was also received by others such as Richard P. Feynman, Nobel Prize in Physics, and John Milnor, Fields Medal. In 1950 he read his thesis at Princeton University on properties of certain geometric spaces known as Kähler manifolds. After working as a professor at the University of Minnesota, in 1964 Calabi joined the University of Pennsylvania. A few years later, he obtained the prestigious Thomas A. Scott Professor of Mathematics, which he held until his retirement in 1994, when he became professor emeritus at the same institution.
His work has left a deep mark on modern geometry. His obsession was to give the bare space a “preferred” shape, like someone who molds a piece of clay with his hands in search of a hidden figure, never before imagined. For example, when placing a rope tied at its ends on a flat surface, what is the preferred shape it can take? The answer of many will be a circumference, because it is “the same everywhere” or, perhaps, because it is “the most perfect figure.” A mathematician could add that this perception has to do with a variational property of said curve: it is the one that maximizes the total area it contains inside.
A mathematical method to find these preferred curves in the plane – called flow of average curvature – is the following: we start from any curve (that does not intersect itself) and make it “evolve” in such a way that it loses area to constant speed and its perimeter decreases as quickly as possible. Over time the curve will be convex, and will tend to a circle of smaller and smaller radius until collapsing at a point. Moments before this collapse, the preferred shape of the curve is observed with the naked eye on a very small scale.
If the initial curve were to cut itself, it may develop a singularity or “spike” throughout its evolution and this changes the preferred shape of the curve. By placing oneself in the place of the singularity just before it is formed, one observes, through a change in scale, the “self-similar” evolution of a curve that comes from an infinite past: a curve that does not change its shape while it evolves with the time. In this case, furthermore, the curve moves by translations, that is, all its points move at a constant speed in a fixed direction. Eugenio Calabi discovered this solution to mean curvature flow in the 1980s and named it La Parca (the Grim Reaper).
It seems that Calabi made this discovery during a tea break, in the middle of a conversation, surrounded by his colleagues. The Grim Reaper turns out to be the only solution defined from an infinite past time of the flow of mean curvature that evolves by translations: an essential property that would only be understood many years later. This is possibly one of Calabi’s most unique characteristics: his influence on the work of his colleagues often occurred through long informal conversations, with keen observations and key examples, which would later become fundamental pieces of future mathematical theories. In the words of Edoardo Vesentini (researcher at the Scuola Normale Superiore di Pisa): “in the most intimidating theories and in those theorems that tormented me the most, Calabi’s simple explanations arrived.”
In his case, these explanations seemed to come from an intuition or aesthetic taste. As Calabi himself explained on a visit to Spain in September 2000: “The main source of geometric intuition is, ultimately, linked to our sensory perceptions of the world. Of course, as we get to more abstract areas, one has to interpret what sensory experience means. I have tried to make this as visible as possible to convey this idea.”
The pleasure of pure discovery and the beauty of geometry were, in fact, two driving forces of Calabi’s mathematics. However, his work has turned out to have important implications in other applied fields, such as theoretical physics. As Calabi himself described, mathematicians “invent imaginary worlds, and scientists decide much later whether these can harbor genuine scientific theories.” One of these worlds imagined by Calabi was born from studying the preferred shape of an important class of geometric spaces known as complex manifolds. These objects are made rigid by providing them with a notion of distance (called the Kähler metric). The preferred shape of this space is given by choosing, among all the possible metrics, the one that makes the space curve more homogeneously. A particular case of this problem is known as the Calabi problem. For more than 20 years, great mathematicians tried to address it, arriving at contradictory solutions. Finally, in 1978, Shing-Tung Yau solved it, giving rise to the spaces popularly known as Calabi-Yau varieties. For this important achievement, the international mathematical community awarded the Chinese mathematician the Fields Medal in 1982. To this day, the general problem initially posed by Calabi remains open and has had a great impact on the development of complex geometry in the 20th century. and beginnings of the 21st century. Much of his activity has focused for years on the study of the geometries known as Kähler-Einstein, of which the Calabi-Yau manifolds are a particular case.
Calabi’s criterion for finding the preferred shape of space turned out, years later, to have a deep relationship with the field equations of general relativity introduced by Albert Einstein. In these equations, the distribution of matter and energy in space determines its curvature. In the absence of matter, or when we establish a homogeneous distribution of it, space adopts the preferred form imagined by Eugenio Calabi. Surprisingly, far from being a mere analogy, Calabi-Yau spaces, with their beautiful geometric shapes, play a key role in some modern physical theories that address the problem of quantum gravity, such as the well-known superstring theory.
Mario Garcia Fernandez He is a Ramón y Cajal researcher at the Autonomous University of Madrid and a member of the Institute of Mathematical Sciences (ICMAT).
Oscar García-Prada He is a research professor at the Higher Council for Scientific Research and a member of ICMAT.
Coffee and Theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
Editing and coordination: Agate A. Rudder G Longoria (ICMAT).
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